On the other hand, a pure AdS vacuum is non-linearly unstable and may eventually form a black hole due to turbulent instability [1]}, [2]}. AdS spacetime has a boundary and acts like a confining box. Therefore, any finite excitation in the AdS would not be dissipated and might be expected to explore all possible configurations, which may eventually lead to the formation of small black holes. For example, a rotating black hole is expected to form as a result of gravitational turbulent instability in AdS [2]}. To see if the proposed scenario [4]}, [5]}, [6]} is applicable even to a Kerr-AdS\({}_5\)  black hole, it is important to study if we can construct the solution of an expanding bubble nucleated via the vacuum decay in Kerr-AdS\({}_5\)  spacetime. The nucleation of a vacuum bubble surrounding a black hole has been actively studied (see, e.g., Refs. [7]}, [8]}, [9]}, [10]}, [11]}, [12]}, [13]}, [14]}, [15]}, [16]}), mainly in the motivation of the Higgs metastability [17]}, [18]}.
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